The Dirichlet heat kernel in inner uniform domains: local results, compact domains and non-symmetric forms

Abstract

This paper provides sharp Dirichlet heat kernel estimates in inner uniform domains, including bounded inner uniform domains, in the context of certain (possibly non-symmetric) bilinear forms resembling Dirichlet forms. For instance, the results apply to the Dirichlet heat kernel associated with a uniformly elliptic divergence form operator with symmetric second order part and bounded measurable coefficients in inner uniform domains in $\mathbb R^n$. The results are applicable to any convex domain, to the complement of any convex domain, and to more exotic examples such as the interior and exterior of the snowflake.